Recently, the advent of quantum computers and the invention of new quantum algorithms have provided a novel paradigm for solving PDEs. A cornerstone of many of these quantum algorithms is the seminal Harrow-Hassidim-Lloyd (HHL) algorithm [1]} for solving linear systems, which can be utilized to solve PDEs by discretizing the PDE and mapping it to a system of linear equations. Compared to classical algorithms, the HHL algorithm can be shown to exhibit an exponential speedup. Unfortunately, attractive as it may sound, the HHL algorithm works only in an idealized setting, and a list of caveats must be addressed before it can be used to realize a quantum advantage [2]}. Moreover, implementing HHL and many other quantum algorithms would require the use of a fault-tolerant quantum computer, which may not be available in the near future [3]}. Instead, the machines we have today are imperfect, noisy intermediate-scale quantum (NISQ) devices [4]} with both coherent and incoherent errors limiting practical circuit depths.
